Everyone knows that there are 230 space groups, and these belong to one
of 32 point groups, which, in turn, belong to one of the 14 Bravais
lattices, and 7 crystal systems: triclinic, monoclinic, orthorhombic,
tetragonal, hexagonal, rhombohedral and cubic.
Or are there? If you look in ${CLIBD}/symop.lib of your nearest CCP4
Suite install, you will find not 230 but 266 entries for space groups,
and 43 different kinds of point groups. And those so-called
rhombohedral systems can apparently be represented as hexagonal, so
maybe there are only six crystal systems?
Blasphemy! (I can almost hear the purists now) But, the point I am
trying to make here is that there is a disconnect between the
traditional way that crystallography is taught (aka Chapter 1: crystal
symmetry) and the pragmatic practice of crystallography (aka what
MOSFLM is doing). It is ironic really that the first thing you must
decide for a new crystal is its space group when in reality it is the
last thing you will ever be certain about it. Probably one of the most
common examples of this is the P2221 and P21212 space groups.
Technically, P2122, P2212, etc are NOT space groups! However, given
that orthorhombic unit cells are traditionally sorted abc, simply
giving such a unit cell with the space group P2221 is not enough
information to be sure which axis is the screwy one. Also, I'm sure
many of you have noticed that for any trigonal/hexagonal crystal there
is always a C222 cell that comes up in autoindexing? This is because
you can always index a trigonal lattice along a diagonal and that
makes it look like centered orthorhombic. But, if you try going with
that C222 choice you find that it doesn't merge ... most of the time.
The fact of the matter is that all autoindexing algorithms give you is a
unit cell, and that is just six numbers. The cell dimensions generally
allow you to EXCLUDE a great many symmetry operations, but they can
never really INFER symmetry. Except, of course, in the special case
where all three angles of the reduced cell are not 90 (or 60) degrees,
then the only possible space group is P1. On the other hand, it is
perfectly possible to have P1 symmetry with all three cell edges the
same length and all angles 90 degrees. It just isn't very likely (in
the posterior probability sense). This is why MOSFLM and other
autoindexing programs pick the highest-symmetry lattice and give you a
space group consistent with that lattice, even though there are plenty
of other possibilities. This is why you should always take the space
group that comes out of autoindexing with a grain of salt. Do NOT make
the mistake of classifying your crystals by the result of autoindexing
alone!
Something similar is true for point groups. A high Rsym for a given
symmetry operator (like you will see in the output of pointless) means
that there is NO WAY that the given symmetry operation is part of the
space group. A low Rsym, however, does not mean that you have a given
symmetry. Could always be some kind of twinning or
nearly-crystallographic NCS (NCNCS?). Twinning is relatively rare, and
gets increasingly rare as you get into the non-merohedral stuff, but it
is always a possibility. Yes, intensity statistics can tell you
something is twinned, but if you have just the right mixture of twinning
and pseudotranslation, then the twinning can go undetected. So, in
general, you can always have _less_ symmetry than you think, but proving
the existence of a symmetry operation is hard.
Space groups, or narrowing down the screw vs rotation nature of various
axes generally requires phasing and looking at a map. The one with
right-handed alpha helices is the correct one. Yes, there are plenty of
tricks like systematic absences, native Pattersons and the like but
there are a lot of false positives and false negatives possible with
each of these. In fact, you tend to throw out more rejects in scaling
than you ever have observations of systematic absences, so why trust
those absent spots so much? In fact, sometimes you need to even go
all the way to the end of refinement to settle the space group. It is
possible to get stuck with R/Rfree too high because the crystal very
slightly violates the symmetry you think it has. (NCNCS again)
Whatever you do, don't forget to try all the possible P2122-like space
groups if you are searching for heavy atoms or doing MR with a primitive
orthorhombic crystal. Far too many people have missed solving their
structure because they didn't know to do this! Fortunately, modern
computers tend to have 8 or so CPUs in them, and there are never more
than 8 space groups possible on any given point group. So, you might as
well launch 8 parallel MR or heavy-atom site-finding jobs in different
space groups, since it will take just as long to run 8 jobs as it will
take to do only one. Well, okay, some of the non-protein ones have more
than 8
